| L(s) = 1 | − 2.56·2-s + 2.54·3-s + 4.56·4-s − 3.00·5-s − 6.50·6-s − 3.61·7-s − 6.56·8-s + 3.45·9-s + 7.70·10-s + 2.20·11-s + 11.5·12-s − 3.81·13-s + 9.26·14-s − 7.64·15-s + 7.68·16-s − 7.79·17-s − 8.86·18-s + 1.99·19-s − 13.7·20-s − 9.19·21-s − 5.64·22-s + 2.98·23-s − 16.6·24-s + 4.03·25-s + 9.78·26-s + 1.16·27-s − 16.4·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 1.46·3-s + 2.28·4-s − 1.34·5-s − 2.65·6-s − 1.36·7-s − 2.31·8-s + 1.15·9-s + 2.43·10-s + 0.664·11-s + 3.34·12-s − 1.05·13-s + 2.47·14-s − 1.97·15-s + 1.92·16-s − 1.89·17-s − 2.08·18-s + 0.458·19-s − 3.06·20-s − 2.00·21-s − 1.20·22-s + 0.622·23-s − 3.40·24-s + 0.807·25-s + 1.91·26-s + 0.224·27-s − 3.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3434077004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3434077004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 17 | \( 1 + 7.79T + 17T^{2} \) |
| 19 | \( 1 - 1.99T + 19T^{2} \) |
| 23 | \( 1 - 2.98T + 23T^{2} \) |
| 29 | \( 1 - 8.31T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 7.38T + 53T^{2} \) |
| 59 | \( 1 + 8.73T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 + 9.40T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.054265181771926822619618932350, −7.33369945589521981942249513349, −6.82801850831453604684000290156, −6.55575888371753955020342622735, −4.84006531434401359711238047715, −3.83620026868898041942080427333, −3.18332779121888085762278648215, −2.62662047606694653980950040228, −1.72459769708412942754152004677, −0.33640129832411884228889218871,
0.33640129832411884228889218871, 1.72459769708412942754152004677, 2.62662047606694653980950040228, 3.18332779121888085762278648215, 3.83620026868898041942080427333, 4.84006531434401359711238047715, 6.55575888371753955020342622735, 6.82801850831453604684000290156, 7.33369945589521981942249513349, 8.054265181771926822619618932350
